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Arithmetic Circuits, Monomial Algebras and Finite Automata

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Mathematical Foundations of Computer Science 2009 (MFCS 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5734))

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Abstract

We study lower bounds for circuit and branching program size over monomial algebras both in the noncommutative and commutative setting. Our main tool is automata theory and the main results are:

  • An extension of Nisan’s noncommutative algebraic branching program size lower bounds [N91] over the free noncommutative ring

    \({\mathbb F}\langle{x_1,x_2,\cdots,x_n}\rangle\) to similar lower bounds over the noncommutative monomial algebras \({\ensuremath{\mathbb{F}}}\langle{x_1,x_2,\cdots,x_n}\rangle/I\) for a monomial ideal I generated by subexponential number of monomials.

  • An extension of the exponential size lower bounds for monotone commutative circuits [JS82] computing the Permanent in ℚ[x 11,x 12, ⋯ ,x nn ] to an exponential lower bound for monotone commutative circuits computing the Permanent in any monomial algebra ℚ[x 11,x 12, ⋯ ,x nn ]/I such that the monomial ideal I is generated by o(n/logn) monomials.

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Authors and Affiliations

  1. Institute of Mathematical Sciences, C.I.T Campus, Chennai, 600 113, India

    Vikraman Arvind & Pushkar S. Joglekar

Authors
  1. Vikraman Arvind
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  2. Pushkar S. Joglekar
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Faculty of Mathematics, Physics and Informatics, Comenius University, Research partially funded by grant APVV 0433-06, Bratislava, Slovakia

    Rastislav Královič

  2. Members of EACSL Jury for the Ackermann Award, Poland

    Damian Niwiński

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Arvind, V., Joglekar, P.S. (2009). Arithmetic Circuits, Monomial Algebras and Finite Automata. In: Královič, R., Niwiński, D. (eds) Mathematical Foundations of Computer Science 2009. MFCS 2009. Lecture Notes in Computer Science, vol 5734. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03816-7_8

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  • DOI: https://doi.org/10.1007/978-3-642-03816-7_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03815-0

  • Online ISBN: 978-3-642-03816-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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